Linear Algebra 1 and 2, Algebra 1.
Recommended: Algebra 2.
Representation theory is about understanding and exploiting symmetry using linear algebra. The central objects of study are linear actions of groups on vector spaces. This gives rise to a very structured and beautiful theory, which can be viewed as a generalisation of Fourier analysis to a non-commutative setting.
Representation theory plays a major role in mathematics and physics. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles.
After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group. One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers.
- Recall basics of ring theory.
- Basics in the theory of modules.
- Basics in homological algebra, Wedderburn--Artin and Maschke's theorem.
- The basic theory of group representations, characters and character tables.
- Burnside's theorem.
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Mode of instruction
Lectures, tutorials and assessed homework.
The final grade consists of homework (20%) and a written (retake) exam (80%). To pass the course, the grade for the (retake) exam should be at least 5 and the (unrounded) weighted average of the two partial grades at least 5.5. No minimum grade is required for the homework in order to take the exam or to pass the course. The homework counts as a practical and there is no retake for it; it consists of weekly assignments, of which the lowest two grades are dropped.
Hendrik Lenstra: Representatietheorie, 2003. Lecture notes by Jeanine Daems and Willem Jan Palenstijn (in Dutch). There is an English translation by Gabriele Dalla Torre.
Serge Lang, Algebra. Springer--Verlag, 2002. (Chapter XVIII is about representation theory; book freely available through SpringerLink)
Jean--Pierre Serre, Linear Representations of Finite Groups. Springer New York, 1977. (Freely available through SpringerLink.)
William Fulton, Joe Harris, Representation Theory: A First Course. Springer New York, 2004. (Freely available through SpringerLink.)
Joseph Rotmanm, Advanced Modern Algebra, Part 2. American Mathematical Society, 2017.
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